Metric Definition and 1000 Threads

Hello, I don't know whether I have mentioned the subject line properly. Many times while reading over General Relativity I come across the following equation: ds^2=dx1^2+dx2^2+dx3^2+dx4^2 =dx^2+dy^2+dz^2-c^2dt^2. Now, my question from the above equation is: (a) Are we putting...

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature) ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j Now the static condition. If I'm not mistaken...

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...

I'm confused about the the definition of a function not being continuous. Is it correct to say f(x) is not continuous at x in the metric space (X,d) if \existsε>0 such that \forall\delta there exists a y in X such that d(x,y)<\delta implies d(f(x),f(y))>ε Is y dependant on \delta? It...

There seems to be an emphasis in several books on general relativity that the metric (components) in itself does not reflect anything physical, only our choice of coordinates. On the other hand it can seem like the authors, instead of being true to this, treat the metric (components) as...

I must be overlooking something! Given a metric space (E,d), the improper subset E is open in E. How? Here is my understanding: 1) We call a set S(subset of E) open iff for all x(element of S) there exist epsilon such that an open ball of radi epsilon centered about s is wholly contained in...

Hi guys. This question is related to Problem 6.3 in Wald which involves deriving the Reissner-Nordstrom (RN) metric. We start with the source free Maxwell's equations ##\nabla^{a}F_{ab} = 0,\nabla_{[a}F_{bc]} = 0## in a static spherically symmetric space-time which, in the coordinates adapted to...

Homework Statement Show that observers whose world-lines are the curves along which only ξ varies undergo constant proper acceleration with invariant magnitude ημν(duμ/dτ )(duν/dτ ) = 1/χ2. Homework Equations ds2 = −a2χ2dξ2 + dχ2 + dy2 + dz2 xμ = {ξ, χ, y, z} t = χsinh(aξ) x...

Thanks in advance - this problem has been bothering me for a while! I'm working with an unpowered spaceship orbiting a large mass M. The orbit is circular and it is following the geodesic freely. It has an orbit radies of r = R. My question is this. The metric of the space-time curvature...

Homework Statement "Evaluate: g^{\mu \nu} g_{\nu \rho} where ds^2 = g_{\mu \nu} dx^\mu dx^\nu , ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 " Homework Equations None necessary, just a notation issue The Attempt at a Solution Just using raising and lowering rules, I imagine each raising and...

Looking for clarification on something. Take the metric in this form ds^2 = dt^2 - dx^2 Brian Green likes to say a null path is one where motion through space is shared equally with motion through time, or (ds=0). Motion through space is not allowed that is "faster" than motion...

hi , how i can demonstrate the expression of metric kerr?

How does one go about finding what the gravitational time dilation is from the metric? Is it simply t'/t_0=1/\sqrt{g_{tt}}? It seems that could be true for static metrics, but perhaps not more dynamic ones like the Kerr metric. My confusion on this arises on how to treat the time cross terms...

hello, Do I have the right to perform the following : gjo,i + g0i,j = (gj0 δij + g0i ),j = (2 g0i ),j Thank you, Clear skies,

Homework Statement If S is a subspace of the metric space X prove (intxA)\capS\subsetints(A\capS) where A is an element of ΩX(Open subsets of X) The Attempt at a Solution So intxA=\bigcupBd(a,r) where d is the metric on X and the a's are elements of A and I think...

Homework Statement A dissimilarity measure d(x, y) for two data points x and y typically satisfy the following three properties: 1. d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y 2. d(x, y) = d(y, x) 3. d(x,z) ≤ d(x, y) + d(y,z)The following method has been proposed for measuring the...

Hi everyone, I posted this a couple days ago and didn't get a response, so I thought I'd try again. Let me know if something about this is confusing. Thanks! Homework Statement Let X be a metric space and let x\in{X} be any point. Prove that the set \left\{x\right\} is closed in X...

Suppose, I have the next metric: g = du^1 \otimes du^1 - du^2 \otimes du^2 And I want to calculate g(W,W), where for example W=\partial_1 + \partial_2 How would I calculate it? Thanks.

Hello all, I've been puzzled by this problem for some time now and was wondering if anyone here could help me out. Textbooks on GR (specifically when going into gravitational waves) tend not to elucidate this. It's often taken for granted that through the gauge diffeomorphism invariance (or...

Given the proposition: A subset U of a metric space is an open set iff U is a union of open balls. Then we are told that it follows from the above proposition that two metrics are equivalent if each open ball of either of the metric topolgies is an open set of the other topology. This is...

a metric on the plane determines an action of SO(2) on is unit circle by rotation. Suppose one starts with a free transitive action of SO(2) on a circle. When does this come from a metric? Always?

How to calculate something relating to the determinant of metric tensor? for example, its derivative ∂_{λ}g. and how to calculate1/g* ∂_{λ}g, which is from (3.33) in the book Spacetime and Geometry, in which the author says that it can be related to the Christoffel connection.

So I ran into a question; Show that there is no metric on S^2 having curvature bounded above by 0 and no metric on surface of genus g which is bounded below by 0. honestly I have no idea what is going on here. I know that a Genus is the number of holes in some manifold or the number of...

Hi can anyone explain how to find \delta \sqrt{-g} when varying with respect to the metric tensor g^{\mu\nu}. i.e why is it equal to \delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g}g_{\mu\nu} \delta g^{\mu \nu}

If you are given a metric gαβ and you are asked to find if it describes a flat space, is there any way to answer it without calculating the Riemman Tensor Rλμνσ? and how can I find for that given metric the coordinate transformation which brings it in conformal form? For example I'll give you...

Metric Space and Topology HW help! Let X be a metric space and let (sn )n be a sequence whose terms are in X. We say that (sn )n converges to s \ni X if \forall \epsilon > 0 \exists N \forall n ≥ N : d(sn,s) < \epsilon For n ≥ 1, let jn = 2[(5^(n) - 5^(n-1))/4]. (Convince yourself...

Homework Statement X is a metric and E is a subspace of X (E\subsetX) The boundary ∂E of a set E is defined to be the set of points adherent to both E and the complement of E, ∂E=\overline{E}\cap(\overline{X\E}) (ignore the red color, i can't get it out) Show that E is open if and only...

So I was flipping through Lang's Undergraduate Analysis and noticed the absence of the important concept of metric spaces. I checked the index and was referred to problem 2, Chapter 6 Section 2. There he defines a metric space, what a bounded metric is, and gives a few straightforward problems...

Are the axioms of a Norm different from those of a Metric? For instance Wikipedia says: a NORM is a function p: V → R s.t. V is a Vector Space, with the following properties: For all a ∈ F and all u, v ∈ V, p(av) = |a| p(v), (positive homogeneity or positive scalability). p(u + v) ≤ p(u)...

could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. usual) metric and makes this open interval into a complete set? Thanks

Am i correct in thinking that in a simple ds^2 =dt^2 - a^{2}d\underline{x}^{2} metric that: g_{\mu\nu}=diag(1,-a^2,-a^2,-a^2) and g^{\mu\nu}=diag(1,-\frac{1}{a^2},-\frac{1}{a^2},-\frac{1}{a^2}) thx

Homework Statement Let X be an infinite set. For p\in X and q\in X, d(p,q)=1 for p\neq q and d(p,q)=0 for p=q Prove that this is a metric. Find all open subsets of X with this metric. Find all closed subsets of X with this metric. Homework Equations NA The Attempt at a...

Hi does anyone know how to calculate: \delta (det|g_{\mu\nu}|) or simply \delta g

Homework Statement Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. Homework Equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists...

We are stating with equivalence principle that passing locally to non inertial frame would be analogous to the presence of gravitational field at that point, so g^'_{ij}=A g_{nm} A^{-1} where g' is the galilean metric and g is the metric in curved space, and A is the transformation which...

my exploration of relativity followed by first reading various books which failed to explain to me how relativity worked but built a strong feel of how one can think about it. after which i decided to take the mathematical way of understanding it for which i am going on with the prof susskind's...

If every continuous function on M is bounded, what does this mean? I am not sure what this function actually is... is it a mapping from M -> M or some other mapping? Is the image of the function in M? Any help would be greatly appreciated!

In Topology: is the multiplicative inverse of a metric, a metric? How do we define the Schwarz inequality then? if ##d(x,z) ≤ d(x,y) + d(y,z)## the inverse ##1/d(x,z)## would give the opposite?

Homework Statement can someone help me to solve these problems in details?? Consider A =(0,1)× R. Is A open w.r.t. the topology induced by the French railway metric in R2? how about B=(-1,1)× R? 2. The attempt at a solution I know A is open in the topology induced by d if and only...

Homework Statement Find the non zero Christoffel symbols of the following metric ds^2 = -dt^2 + \frac{a(t)^2}{(1+\frac{k}{4}(x^2+y^2+z^2))^2} (dx^2 + dy^2 + dz^2 ) and find the non zero Christoffel symbols and Ricci tensor coefficients when k = 0 Homework Equations The...

Could somebody please explain something regarding the Nordstrom metric? In particular, I am referring to the last part of question 3 on this sheet -- http://www.hep.man.ac.uk/u/pilaftsi/GR/example3.pdf about the freely falling massive bodies. My thoughts: The gravitational effects...

Hello, I was wondering if it was possible (or advisable) to read Chapter 7 of Munkres (Complete Metric Spaces and Function Spaces) without having done Tietze Extension Theorem, the Imbeddings of Manifolds section, the entirety of Chapter 5 (Tychonoff Theorem) and the entirety of Chapter 6...

In Minkowski spactime (Flat), if the coordinate system makes a rotation e.g. around y-axis (centred) , for the metric ds^2, how to make the tertad (flat spacetime) as the coordinate system rotats?

Hello Everybody, Instead of solving the geodesic equations for the Schwarzschild metric, in many books (nearly in all books that I consulted), conserved quantities are looked at instead. So take for eg. Carroll, he looks at the killing equation and extracts the equation K_\mu...

I think I remember reading somewhere that all the machinery of manifolds and a metric needed to be established first before the integral and the differential of calculus had any meaning. Am I remembering wrong? Is there such a thing as coordinate independent integration or differentiation? Thanks.

How can find components of tetrads from metric ? i know the relation between tetrads and metric g_{μ \nu}=η_{ab}e^{a}_{μ}e^{b}_{\nu} where e^{b}_{\nu} are component of tetrads , in the case of Schwarzschild that metric is diagonal , it is a easy problem but what about non-diagonal metric like...

Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should...

The question comes from the Munkres text, p. 133 #3. Let Xn be a metric space with metric dn, for n ε Z+. Part (a) defines a metric by the equation ρ(x,y)=max{d1(x,y),...,dn(x,y)}. Then, the problem askes to show that ρ is a metric for the product space X1 x ... x Xn. When I originally...

Hello, I have read that, in a freely-falling frame, the metric/ interval will be of the form: ds2 = -c2dt2(1 + R0i0jxixj) - 2cdtdxi(\frac{2}{3} R0jikxjxk) + (dxidxj(δij - \frac{1}{3} Rikjlxkxl) to second order. Does anyone know where I could find a derivation of this result?

Let $A\subset X$ for $(X,d)$ metric space, then prove that $\text{diam}(A)=\text{diam}(\overline A).$ I know that $\text{diam}(A)=\displaystyle\sup_{x,y\in A}d(x,y),$ but I don't see how to start the proof. The thing I have is to let $\text{diam}(\overline A)=\displaystyle\sup_{x,y\in \overline...